Doesn’t matter for the issue at hand, that’s just a question of language relating to the example. A different example:
“A set always has a maximal element under the larger-than relation for numbers”
“That’s wrong”
“Ah but any set of natural numbers has a maximal element, that is also a set, gotcha”
“No, you just said set, that’s too generic, if you meant any set of natural numbers you should have said that.”
“You’re as stubborn as my brother”
“But your younger brother isn’t stubborn at all”
“I was talking about my older brother”
The average is a generic concept covering multiple more specific concepts like mean and median. If you say something about the generic concept it should not depend on any properties of just one of the specific concepts, in order to hold generally.
Your brother is a term for a single person that is simply under-determined and could turn out to apply to either one, but not both. What you say about your brother should apply to the brother you mean, in order to hold.